Classifying representations by way of Grassmannians

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed dimension $d$ and fixed squarefree top $T$. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations. In case of existence of a moduli space -- unexpectedly frequent in light of the stringency of fine classification -- this space is always projective and, in fact, arises as a closed subvariety ${\mathfrak{Grass}}^T_d$ of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety ${\mathfrak{Grass}}^T_d$ is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of `finite local representation type at a given simple $T$', the radical layering $\bigl( J^lM/ J^{l+1}M \bigr)_{l \ge 0}$ is shown to be a classifying invariant for the modules with top $T$. This relies on the following general fact obtained as a byproduct: Proper degenerations of a local module $M$ never have the same radical layering as $M$

The possible values of critical points between varieties of lattices

We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of W contains a prime interval, we prove that there exists a bounded lattice A in V with at most aleph 2 elements such that Conc(A) is not isomorphic to Conc(B) for any B in W. The bound aleph 2 is optimal. As a corollary of our results, there are continuum many congruence classes of locally finite varieties of (bounded) modular lattices

Infinite combinatorial issues raised by lifting problems in universal algebra

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (k,r,l){\to}m (for proving that a critical point is small). In the middle, we find l-lifters of posets and the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.Comment: 22 pages. Order, to appea